Broadband velocity modulation spectroscopy of ThF+ for use in a measurement of theelectron electric dipole moment

Daniel N. Gresha,∗, Kevin C. Cossela,1, Yan Zhoua, Jun Yea, Eric A. Cornella,

aJILA, National Institute of Standards and Technology and University of Colorado, andDepartment of Physics University of Colorado, 440 UCB, Boulder, CO 80309, USA

Abstract

A number of extensions to the Standard Model of particle physics predict a permanent electric dipole moment of the electron(eEDM) in the range of the current experimental limits. Trapped ThF+ will be used in a forthcoming generation of the JILA eEDMexperiment. Here, we present extensive survey spectroscopy of ThF+ in the 700 - 1000 nm spectral region, with the 700 - 900 nmrange fully covered using frequency comb velocity modulation spectroscopy. We have determined that the ThF+ electronic groundstate is X 3∆1, which is the eEDM-sensitive state. In addition, we report high-precision rotational and vibrational constants for 14ThF+ electronic states, including excited states that can be used to transfer and readout population in the eEDM experiment.

Keywords: Thorium Fluoride, Electron EDM, Frequency Comb

1. Introduction

A measurement of the permanent electric dipole moment ofthe electron (eEDM) is a direct test of parity- and time-reversalsymmetry violation [1, 2, 3]. Electron EDM experiments havetypically been performed on beams of neutral atoms [4] or neu-tral molecules [5, 6], with the current experimental limit of|de| < 9.7×10−29 e·cm set by the ACME collaboration’s molec-ular beam experiment [7]. The JILA eEDM experiment in-stead uses trapped molecular ions, leveraging the large effectiveelectric field [8, 9] and suppression of eEDM systematic errorsavailable in molecular 3∆1 states [10, 11], while also achievinga long measurement coherence time [12]. The JILA experimentis currently using trapped HfF+ [13], but ThF+ is a promisingcandidate molecule for a future generation experiment due toits larger effective electric field [14, 15, 16] and longer-lived3∆1 state. Assigning the ground state of ThF+ has been exper-imentally challenging due to the proximity of both 1Σ+ and 3∆

states [17, 18]. Calculations suggested that 3∆1 might be theground state but uncertainties prevented a conclusive determi-nation [19, 15]. Here, we show that the ThF+ ground state isX 3∆1, and that the a 1Σ+ state lies 314 cm−1 higher in energy.This is beneficial for the eEDM experiment, since the measure-ment coherence time will not be limited by the spontaneouslifetime of the “science” state. In addition to 3∆1 and 1Σ+, wehave measured high-precision rotational and vibrational con-stants for 12 additional electronic states. The results reportedhere are benchmarks for ab initio theory, and may assist in un-derstanding actinide bonding.

∗Corresponding AuthorEmail address: dgresh@jila.colorado.edu (Daniel N. Gresh)

1Current address: National Institute of Standards and Technology, 325Broadway, Boulder, CO

The lack of spectroscopic information about electronic statesof ThF+ above 10,000 cm−1, and the difficulty of perform-ing survey spectroscopy on a small number of trapped ions,necessitated a tool that provides broad bandwidth, high pre-cision, high sensitivity, and ion-selectivity. Frequency-combvelocity-modulation spectroscopy (comb-vms) [20, 21], whichcombines cavity-enhanced direct frequency comb spectroscopy[22, 23] and velocity-modulation spectroscopy (vms) [24, 25],fulfills these requirements. With this system we have scannedthe entire wavelength range of 696 - 900 nm, acquiring 3,260cm−1 of continuous spectra at 150 MHz (0.005 cm−1) step-size.Additionally, guided by the results of the comb-vms and thePFI-ZEKE spectra from Barker et al. [17], we have performedtargeted cw-laser vms scans to identify additional experimen-tally useful transitions and to assign electronic-state Ω valuesvia observation of low-J rotational lines. In total, we have fit27 rovibronic transitions in ThF+ and 5 transitions in a differ-ent thoriated molecular ion, potentially ThO+. We discuss theproduction and detection of ThF+, details of the ground stateassignment, and the assignment of other ThF+ electronic states.

2. Experimental details

2.1. Frequency comb vms

Fig. 1 shows an overview of the comb-vms system. A 3-GHz repetition-rate titanium-doped sapphire frequency combis spectrally broadened in nonlinear fiber and coupled into anenhancement cavity surrounding a 1-kW electrical dischargecell. A VIPA imaging system [23, 27] resolves individualcomb modes and provides 1,500 simultaneously resolved combmodes per image.

The spectroscopically usable fundamental bandwidth of thecomb is 780-850 nm, centered at 815 nm. To obtain maximum

Preprint submitted to September 15, 2015

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Comb

Cavity. . . . . . ... . . . . . .

frep

920 ºCThF4

3 GHz Ti:Sapph Comb+ ber broadening

VIPA dispersion +lock-in cameraFrequency

Figure 1: Experimental schematic. A 3 GHz Ti:Sapph comb is broadenedin nonlinear fiber and coupled into a bow-tie ring cavity, propagating in eitherdirection through a discharge of ThF4. Four liquid crystal variable retardersand a polarizing beam splitter control the direction of light propagation throughthe discharge. The cavity free spectral range is 1/25 of the comb repetitionfrequency. After exiting the cavity, light is sent to a cross-dispersive VIPAimaging system with single comb mode resolution and imaged onto a Heliotislock-in camera [26]. Approximately 1,500 comb modes are resolved in a singleimage.

power throughput to the experiment, the output of the frequencycomb is immediately coupled into the nonlinear fiber to broadenthe spectrum. A half-wave plate before the nonlinear fiber in-put enables control of the output spectrum [28]. No single fibergave a smooth, high power spectrum in the entire 700 - 900nm range, so we used three different fibers, replacing them asneeded to cover the entire region. A Newport SCG-800-CARSfiber resulted in significant power in the 850 - 900 nm spec-tral region but practically no light in the 700 - 780 nm region.85 cm of Thorlabs NL-2.8-850-02 fiber, pumped in the normaldispersion regime, provided a small amount of broadening andresulted in high power in the 730 - 780 nm range. 80 cm ofThorlabs NL-2.4-800 fiber, pumped close to the zero dispersionwavelength, resulted in most of the laser power being shifted tothe 680 - 730 nm regime.

The fiber output spectrum often contained a significant frac-tion of laser power shifted to a spectral region outside of thedesired scanning range. After the fiber, low- and high-pass op-tical filters were used to select a 50 nm spectral window. Thespectrum after the filters was optimized and monitored usinga grating spectrometer with a USB readout. A linear polarizerwas placed after the filters to eliminate the different polariza-tion components due to nonlinear polarization rotation in the

fiber. An electro-optic modulator, driven at 17 MHz, appliesRF sidebands to each comb tooth.

After the modulator, the comb light is fiber-coupled to thetable supporting the discharge cell. A 2.5-meter-long low-finesse (F ≈ 70-100) enhancement cavity surrounds the dis-charge tube. Every 25th cavity mode is coupled to a combmode, and the cavity is locked to the comb using the Pound-Drever-Hall technique[29]. In order to obtain maximum cav-ity transmission, AR-coated windows were placed on each endof the discharge apparatus (R < 0.1% per surface 700 - 900nm), and T = 3% cavity input couplers were used, runningthe cavity in the over-coupled regime. Liquid-crystal variableretarders and a polarizing beam splitter were used to rapidlyswitch the direction of light propagation through the cavity andthe discharge tube. By subtracting the spectrum acquired fromeach direction, common-mode laser noise and asymmetricaldischarge effects, such as an asymmetrical concentration mod-ulation of molecules, are suppressed.

After exiting the cavity, the comb light is imaged using a 2DVIPA spectrometer [30] with single comb mode resolution ontoa (Heliotis) lock-in camera [26]. To fully resolve different VIPAorders, a 1800 lines/mm diffraction grating was used for the 780- 900 nm spectral range, and a 2160 lines/mm grating was usedfor the 700 - 780 nm range. A cw titanium-doped sapphire laseris co-propagated with the comb light for frequency calibra-tion. The cw laser is locked to its internal reference cavity, andthe frequency is measured by a 100-MHz-accuracy wavemeter.With interleaving scans, we obtain a maximum of 150 cm−1 ofcontinuous spectra over 1,500 simultaneous spectral channels at150 MHz (0.005 cm−1) step-size in a 25 minute total data col-lection time, with 3 × 10−8 Hz−1/2 (spectral element)−1/2 frac-tional absorption sensitivity. The acquisition time is currentlylimited by software and with improvements could be shortenedto 10 minutes.

The discharge tube is a straight 50 cm length of 1.7 cm inner-diameter, 1.9 cm outer-diameter quartz tubing. The tube is cen-tered inside a home-built oven, which is heated to 920 C toobtain a sufficient vapor pressure of ThF4 [31, 32]. Initially,smaller diameters of quartz tubing were used but were rapidlyblocked during operation. The blockage is likely due to the re-action of ThF4 with quartz at high temperature to form ThO2[33], which has a low vapor pressure at 920 C [34] and shouldrapidly plate out along the discharge tube. A straight alumina(Al2O3) tube was tested but the discharge ceased to run downthe length of the tube above 800 C, instead opting to dischargeto the grounded gas lines.

For each oven run, approximately 1g of ThF4 is loaded intothe discharge tube, and the tube is pumped out with a scrollpump to 10 mTorr. Helium gas flows through the tube at a pres-sure of 5 Torr. Three heaters are used to maintain a temperatureof 920 C in the center of the oven and a temperature of 880C on either edge of the oven. Higher temperatures were foundto deplete the ThF4 faster but not to yield a significant increasein ThF+ signal. Single oven runtimes were typically 3-4 hoursbefore the signal fell to 80% of its peak value due to depletionof ThF4.

During some scans we observed bands that we tentatively

2

attribute to ThO+ (see Section 3.3). Attempts were made toremove the ThO+ signal from the discharge while monitoringThF+ and ThO+ absorption strengths with cw-vms (see nextsection). Running the discharge with different buffer gases,such as Ne or Ar, resulted in the loss of all ThF+ signal whileretaining the same ThO+ signal. Contrary to our expectations,seeding the discharge with SF6 gas again resulted in the lossof all ThF+ signal and doubled the ThO+ line strengths. In allcases switching back to a He discharge resulted in an immedi-ate return to prior ThF+ line strengths. The ThF+ signal wasonly positively affected by total oven runtime,

i.e. time when the oven was at 920 C with ThF4 loadedand a discharge running. After roughly 5 total hours of ovenruntime, the ThO+ signal decreased by a factor of two and theThF+ signal increased by a factor of two. This is likely dueto Th-based discharge byproducts coating the tube walls andsuppressing the ThF4 – quartz reaction.

2.2. cw Ti:Sapph and diode laser vmsTwenty-one vibrational bands were fit in the comb-vms spec-

trum. The total spectral coverage of the comb-vms system islimited by coatings on the VIPA etalon and by the decreasedquantum efficiency of the silicon-based lock-in camera at wave-lengths longer than 900 nm. Cw-laser vms was used to extendspectral coverage beyond the 700 - 900 nm range in order toidentify additional favorable electronic states for coherent pop-ulation transfer between X-a and to increase the signal-to-noiseratio on low-J rotational lines. For example, low-J lines of theΩ = 0−← 3∆1 (0, 0) band were observed at 685.4 nm, a wave-length inaccessible with the current optics coatings in the comb-vms system (see Section 3.1).

Due to the lower complexity of cw scans, a higher single-pass signal-to-noise ratio can be obtained. However, the totaldata acquisition time is more than 20 times slower per band-width than comb-vms. For the cw scans, the enhancement cav-ity was removed, and the laser was split into two componentswith a power ratio of approximately 2:1 that counterpropagatedthrough the discharge tube. Roughly 1 mW of power is con-tained in the lower power beam. The lock-in signal is detectedon the difference port of an autobalancing detector [35]. Thebest scan rate achieved is 1 cm−1 in 4 minutes with a 300 mslock-in time constant, limited by reading the wavemeter andscanning slowly enough to maintain laser lock. The best single-pass fractional sensitivity reached is 2 × 10−8 Hz−1/2, about afactor of 2 above shot noise at 1 mW, limited by a combinationof electrical and acoustic pickup from the discharge.

Low-J lines of the Ω = 0− ← 3∆1 (0, 0) band at 685.4 nmwere scanned using an external-cavity diode-laser with a 680nm diode. The high-precision rotational and vibrational con-stants measured from the Ω = 0− ← 3∆1 (0, 1), (1, 2), (2, 3),(0, 2), and (1, 3) bands allowed us to pinpoint line centers inthe (0, 0) vibrational band to 0.1 cm−1 accuracy. Feed forwardto the diode current resulted in a mode-hop free scan range of0.2 cm−1. Fractional sensitivity with 50 µW optical power waslimited by electrical pickup to 1× 10−7 Hz−1/2, yielding signal-to-noise > 10 on the weakest low-J lines with a 1 s lock-in timeconstant.

3. Results and analysis

We have observed 14 different ThF+ electronic states. To in-dicate state assignments Hund’s case (a) notation is used whenpossible, and case (c) notation is used when only the Ω-value isknown. All of the acquired data is shown in Fig. 2 (a): a totalof 3,260 cm−1 was acquired with comb-vms (black), and 400cm−1 with cw-vms (red). ThF+ bands often presented with atleast a single P-, Q-, and R-branch, and frequently had two ofeach branch due to Λ-doubling in both electronic states. Fig.2 (b) shows a typical six branch vibrational band, with a clearR-branch band head and a strong Q-branch. The Λ-doubling isstrong enough to fully resolve the two parity components by J= 3.

For an observed vibrational band, rotational constants are de-termined by fitting assigned line centers, defined as the zero-crossing, to a general energy expression:

E = ν0 + (B′ − s′k′/2)J′(J′ + 1) − (D′ − s′k′D/2)J′2(J′ + 1)2

− [(B′′ − s′′k′′/2)J′′(J′′ + 1) − (D′′ − s′′k′′D/2)J′′2(J′′ + 1)2].(1)

Here, s is a parity term: s = +1 for e-symmetry and s = −1 forf -symmetry, and k and kD are generic Λ-doubling constants pro-portional to J(J+1) and J2(J+1)2, respectively. Bands with Λ-doubling observed in each electronic state have all six branchesfit simultaneously, with s = ±1 corresponding to the two paritylevels in each state. Most observed transitions have the promi-nent Λ-doubling contribution arising from the upper state; inthese transitions k′′ and k′′D are set to 0.

Vibrational constants are determined from the relationship,

νv′v′′ = T ′e − T ′′e + ω′e(v′ + 1/2) − ωeχ′e(v′ + 1/2)2

− [ω′′e (v′ + 1/2) − ωeχ′′e (v′ + 1/2)2],

(2)

where νv′v′′ is the (v′, v′′) band origin, ωe is the vibrational con-stant, ωeχe is the anharmonicity constant, and Te is the energyto the minimum of the electronic potential. For states with toofew transitions observed to directly determine ωe and ωeχe, aMorse potential was assumed, allowing the substitution

ωeχe =α2

eω2e

36B3e

+αeωe

3Be+ Be, (3)

where αe is the rotational-vibrational coupling constant and Be

is the equilibrium rotational constant.Vibrational assignments are determined by comparing ob-

served band intensities to Franck-Condon factors and a thermalvibrational distribution. With prior knowledge of vibrationalconstants from Barker et al. [17], and our measured B val-ues, Franck-Condon factors were calculated and compared toobserved intensities for the initial bands measured. With therotational and vibrational constants we have observed in ThF+,the vibrational (0, 0) is always the strongest band. The highestintensity bands in the ∆v = ±1 vibrational series are about 1.5-2times weaker than the (0, 0) band.

Due to the high density of electronic states, the relativelysmall difference in vibrational constants for different electronic

3

Frequency (cm-1)

Frac

tiona

l abs

orpt

ion

(x10

-5)

10,000-6

-4

-2

0

2

4

6

10,500 11,000 11,500 12,000 12,500 13,000 13,500 14,000 14,500

11,936 11,938 11,940 11,942 11,944 11,946

-1

-0.5

0

0.5

1

-1

-0.5

0.5

-1.5

0

1.5

1

14,588.5 14,589 14,589.5 14,590 14,590.5 14,591

(a)

(b) (c)

E (cm-1)

14,591.5

16,682

3Δ1

1Σ+

3Δ2

3Δ3

3Π0

Ω = 0+

Π1

Π1

Ω = 0+Ω = 0-

3Π2

14,58914,29614,224

12,994

10,472

3,3953,150

1,052

3140

Figure 2: All acquired ThF+ spectrum. The energy level diagram on the right indicates all experimentally observed electronic energy levels. All y-axes are in10−5 single pass fractional absorption. All x-axes are in cm−1. (a) All ThF+ spectra acquired with comb-vms (black) and cw-vms (red). (b) Fitted positions toΠ1[12.99] ← 3∆2 (0, 0). Note the presence of six branches – two P- (purple and green), two Q- (dark blue and light blue) and two R- (red and orange) – due toΛ-doubling in each electronic state. (c) Low-J lines in the Ω = 0−←3∆1 (0, 0) band. The solid colors indicate different branches for an Ω′ = 0← Ω′′ = 1 transition(P- green, Q- blue, R- red). The dashed orange lines represent the R-branch intensity if the transition was instead Ω′ = 1 ← Ω′′ = 0. In each case the overall bandintensity is set by scaling the line intensities to match the strongest R-branch lines, which are off-scale peaks marked by solid red lines. P(1) is present, and R(0) isclearly absent.

states, and the high oven temperature, the ThF+ spectrum is of-ten very congested. Fits are performed by manually identifyingbranches and assigning line centers to the rotational structuregiven by Eqn. 1.

We have developed a simple automatic line finder to assistwith identifying bands in the most congested regions. Localmaxima are identified and then matched with the nearest zerocrossing of the correct line-shape phase within the observedlinewidth of 1 GHz. A minimum intensity threshold is specifiedto avoid assigning noise to real lines, and a maximum intensitythreshold is used to avoid assigning strong atomic features tomolecular rotational lines.

The identified lines are plotted in a Loomis-Wood style fig-ure, which displays J = 0-100 residuals on a single figure. Oversuch a large scale, pattern-forming rotational sequences can beeasier to identify. By modifying B values, D values, and theband origin, ν0, it is possible to precisely identify the line cen-ters of hundreds of rotational lines at once. Additionally, thelarge J-range of the plots enables clear identification of smallerhigh-J effects, such as Λ-doubling scaling as J4, which typicallymanifests in the spectrum as lines splitting into doublets above

J = 40.

3.1. 3∆1 ground state assignment

Assignment of the low-lying 3∆1 state is of paramount impor-tance to the eEDM experiment. Initial PFI-ZEKE experimentsfrom Barker et al. [17] observed both the 1Σ+ and 3∆1 states andmeasured the electronic energy difference to be 315(1) cm−1.However, due to the inability to fully resolve low-J lines, com-pounded by the similarity of B and ωe values for the X and astates, it was not possible to conclusively assign either as theThF+ ground state [17]. Our assignment necessitates observa-tion of an excited electronic state with coupling to both the 1Σ+

and 3∆1 states. Such a state will have a spectral signature oftwo vibrational bands with band origins separated by the X-aenergy difference. In order to couple to both lower states, theupper state must be either Ω = 0+ or Ω = 1. The final assign-ment uses measurement of the Λ-doubling and an observationof low-J lines.

In the comb-vms data we have fit transitions at ν0 =

13935.996(7) cm−1 and ν0 = 13286.716(7) cm−1 with one set

4

Frequency (cm-1)

Frac

tiona

l abs

orpt

ion

(x10

-6)

10,468 10,472

E (cm-1)16,682

3Δ1

1Σ+

3Δ2

3Δ3

3Π0

Ω = 0+

Π1

Π1

Ω = 0+Ω = 0-

3Π2

14,58914,29614,22412,994

10,472

3,3953,150

1,052

3140

10,476 10,480

-3

0

3

Frequency (cm-1)

(a)

10,152

-1

0

1

10,156 10,160

(b)

10,164

Figure 3: Spectrum of 3∆1 and 1Σ+ coupling to a common Ω = 0+ state. (a) An Ω = 0+ ← 3∆1 vibrational band is observed at ν0 = 10471.889(7) cm−1 (redaxes). A strong Q-branch (blue markers) is observed in addition to the P- (green markers) and R-branches (red markers). (b) A transition is observed 314 cm−1

lower in energy at ν0 = 10157.607(7) cm−1 (blue axes). This transition has no Q-branch, indicating Ω′′ = Ω′ = 0. The transition ordering indicates that 3∆1 islower in energy as illustrated in the energy level diagram on the right.

of P-, Q-, and R-branches and a nonzero k constant. The pres-ence of a nonzero k and one of each branch implies that the twoelectronic states in the transition must be Ω = 0 and Ω = 1. Thesmall magnitude of k, |k| = 0.869(7)×10−4 cm−1, makes it im-possible to differentiate between k in the upper state, k′, or in thelower state, k′′. Becauseωe for the 1Σ+ and 3∆1 states is so simi-lar it could not be used for electronic state identification. Basedon calculated Franck-Condon factors and the intensity of thethree transitions observed at ν0 = 13935.996(7), 13876.539(8),and 13817.279(8) cm−1, and the two transitions observed at ν0= 13286.716(7) and 13231.099(8) cm−1, the transition at ν0 =

13935.996(7) cm−1 was identified as the Ω = 0−← 3∆1 (0, 1)vibrational band.

The lower state rotational constant, B0 = 0.24264(3) cm−1,and the vibrational constant, ωe = 656.96(1) cm−1, both agreewith Barker et al. [17] for low-lying states of ThF+. Also, theJ2 Λ-doubling proportionality constant, |k| = 0.869(7) × 10−4

cm−1, is consistent with expectations for a ∆Ω=1 state: J2 scalingand small magnitude [36]. Thus, we tentatively assign the lowerstate to 3∆1.

To confirm the lower state Ω = 1 assignment we measuredlow-J lines of the Ω = 0−← 3∆1 (0, 0) band at ν0 = 14589.09(2)cm−1, as shown in Fig. 2 (c). Because of the high oven temper-ature the spectrum is typically congested, but we were able tomeasure low-J lines of this band due to the high line strength.Higher-lying Q- and R-branch lines were scanned to confirmrotational constants and to fix the overall intensity of the band.Two compelling pieces of evidence are seen. First, the R(0)line, indicated by the dashed orange line, is absent, while theP(1) line, indicated by the solid green marker, is present. Weclearly see R(1)-R(4). P(2)-P(4) are covered by strong Q-branch lines. Secondly, the Honl-London factors [37] for anΩ′′ = 0 − Ω′ = 1 transition considerably increase the intensityof the low-J R-branch lines relative to an Ω′′ = 1 − Ω′ = 0transition, indicated by the dashed orange lines and the solidred lines, respectively. The observed R-branch line intensities

agree with Ω′′ = 1. This confirms that the lower state is Ω = 1and the upper state is Ω = 0.

We have observed another transition at ν0 = 10471.889(7)cm−1, with one of each branch and nonzero k, shown in Fig.3 (a). The fitted values of B and D agree with B0 and D0 for3∆1. Additionally, the magnitude of k agrees with the previ-ously measured transition at ν0 = 13935.996(7) cm−1 but thesign is changed, which is expected if the upper state is of a dif-ferent overall parity, e.g. an Ω = 0+ instead of an Ω = 0− state.Therefore, the lower state in this transition is also 3∆1.

A vibrational band with no Q-branch is observed 314.282(7)cm−1 lower in energy, consistent with the X-a separation mea-sured by Barker et al. [17], with ν0 = 10157.607(7) cm−1. Theband origin is shown in Fig. 3 (b). The absence of a Q-branchindicates the two electronic states in the transition both have Ω

= 0, giving strong evidence for the lower state in this transitionbeing 1Σ+. To confirm the ground state assignment, the upperelectronic state must be the same as the one observed in the ν0= 10471.889(7) cm−1 transition.

With no Q-branch the band cannot be fit a priori. How-ever, by fixing B′ and D′ to the values measured in the ν0 =

10471.889(7) cm−1 transition, we obtained a good fit with rea-sonable residuals. The lower-state rotational constant fits toB′′ = 0.24542(3) cm−1. This is larger than the fit B(3∆1), asintuitively expected since an s2 electron configuration (in 1Σ+)is typically more tightly bound than an sd configuration (alsoobserved in HfF+ [21]).

The fit residuals provide the final piece of compelling evi-dence for the upper electronic state being the same. Fit resid-uals for the Ω = 0+[10.47]← 3∆1 (0, 0) at ν0 = 10471.889(7)cm−1 are plotted in Fig. 4 (a). A strong avoided crossing per-turbation is evident at approximately J′ = 62. Other 3∆1 fitshave no avoided crossing, so the perturbation must be presentin the Ω = 0+ upper electronic state. Fit residuals for the ν0 =

10157.607(7) cm−1 transition, shown in Fig. 4 (b), display anavoided crossing of the same magnitude again at J′ = 62. This

5

Ω = 0+[10.47] 3Δ1 (0, 0)

Ω = 0+[10.47] 1Σ+ (0, 0)

Resi

dual

(cm

-1)

Resi

dual

(cm

-1)

0

-0.08

0.04

0.08

-0.04

J’0 20 40 60 80 100

-0.08

0.04

0.08

-0.04

0

J’0 20 40 60 80 100

PQR

PR

(a)

(b)

Figure 4: Fit residuals for 3∆1 and 1Σ+ coupling to a common upper state.The fit residuals of each transition exhibit an avoided crossing perturbation ofthe same magnitude at the same rotational quantum number, confirming theupper electronic state is the same in each transition. The dashed grey lineson each figure are guides to the eye and are identical. (a) Fit residuals forΩ = 0+[10.47]←3∆1 (0, 0). (b) Fit residuals for Ω = 0+[10.47]←1Σ+ (0, 0).

confirms that the upper state is the same for the two transitions,and thus that the assignments of 1Σ+ and 3∆1 are correct. Sincethe Ω = 0+[10.47]← 1Σ+ (0, 0) transition frequency is lowerthan that of the Ω = 0+[10.47]← 3∆1 (0, 0) transition, 3∆1 isthe electronic ground state.

3.2. Additional ThF+ electronic states

In addition to the 1Σ+ and 3∆1 electronic states we have ob-served rovibronic transitions that we have assigned to originatefrom the 3∆2 and 3∆3 states. These are given in Table 1 as theΠ1[12.99] ← 3∆2, the Π1[14.22] ← 3∆2, and the 3Π2 ←

3∆3groups of transitions.

For the 3∆2 state, five vibrational bands of Π1[12.99]← 3∆2and one band of Π1[14.22]← 3∆2 were observed. Each bandhas two of each P-, Q-, and R-branch originating from a com-mon band origin, which indicates that Λ-doubling is presentin each state. A small section of the Π1[12.99] ← 3∆2 (0,0) band is shown in Fig. 2 (b). Two strong Q-branches areseen, indicating that Ω′ , Ω′′. The Λ-doubling has a strongJ′2 dependence, consistent with the upper state being predomi-nantly ΠΩ=1 [38, 36]. B′′e = 0.24342(2), similar to the value ofBe(3∆1) = 0.24311(7), and consistent with the measured valuefor 3∆2 from Barker et al. [17]. Additionally, ω′′e is consistent

with the value measured by Barker et al. and again similar toωe(3∆1).

To confirm this assignment, we observed low-J lines in theΠ1[12.99]← 3∆2 (0, 1), (0, 0), and (1, 0) bands as well as inthe Π1[14.22]← 3∆2 (0, 0) band using the cw Ti:Sapph. In eachcase, Q(1) is absent and Q(2) is present, thus leaving Ω′′ = 1 or2 as the only possibilities. R(1) is absent in each band, and theintensity of the low-J P- and R-branch lines is consistent withΩ′′ > Ω′. Therefore, Ω′′ = 2, Ω′ = 1, and the lower electronicstate is 3∆2.

The assignment for 3∆3 is similar. Three vibrational bandsof 3Π2←

3∆3 were observed, with each band again having twoof each P-, Q-, and R-branch at a common band origin. Herethough the Λ-doubling has a strong J′4 dependence, as expectedfor a 3ΠΩ=2 upper state [38]. B′′e = 0.24388(7) is similar tothat of 3∆1 and 3∆2, and consistent with Barker et al. [17].Scanning near the band origins of the 3Π2 ←

3∆3 (0, 1) and(0, 0) bands with the cw laser revealed that P(2) is absent andP(3) is present. Low-J Q-branch lines could not be resolved dueto contamination from other bands. However, the intensity ofthe low-J P- and R-branch lines is again consistent with Ω′′ >Ω′, concluding that Ω′′ = 3 and Ω′ = 2. This confirms thelower state 3∆3 assignment. The assignment of upper state 3Π2is based on the strong, J′4 Λ-doubling, and the expected strongtransition strength for a 3Π2 ←

3∆3 transition.Several bands exhibit strong perturbations at high-J, which

manifest as fit residuals that deviate from Eqn. 1 at J & 60. Toobtain uniform fit residuals at high-J it was necessary to limitthe range of J-values, or to include additional fit parameters inthe Hamiltonian, such as HJ3(J + 1)3, or in some cases evenhigher powers of J. Higher vibrational levels of the Π1[12.99]state appear to become perturbed, likely as the result of prox-imity to a perturbing state near 14,000 cm−1. This is evident bythree different observations. First, k′D has a strong vibrationaldependence, increasing from 2.1(2) × 10−9 cm−1 in the v′ = 1vibrational state, to 7.7(8) × 10−9 cm−1 in the v′ = 2, and to10.1(7) × 10−9 cm−1 in the v′ = 3, the highest vibrational statewe observe. Secondly, fit residuals deviate from Eqn. 1 at J> 80 for the v′ = 2 and v′ = 3 vibrational states, but not forthe v′ = 0 and v′ = 1. Finally, assuming a Morse potential,ωe and ωeχe can be calculated for both the lower and upperelectronic states using the (0, 1), (0, 0), and (1, 0) bands. How-ever, the origins of the (2, 1) and (3, 2) bands are inconsistentwith the values calculated, indicating that the upper state elec-tronic potential deviates from the Morse approximation beyondv′ = 1. Similarly, k′D is strongly vibrationally dependent in the3Π2←

3∆3 group of transitions, decreasing from 74(4) × 10−9

cm−1 in the v′ = 0, to 41(9) × 10−9 cm−1 in the v′ = 1. Thisis likely due to a high density of unobserved (dark) electronicstates.

Perturbations are also evident in a number of other vibra-tional bands. The Ω = 0+[14.29]← 3∆1 (0, 0) band shows astrong deviation from the expression of Eqn. 1 above J = 50.The (0, 1) band has the same structure in the fit residuals. The(1, 1) band, which ordinarily would be strong, is not seen, andthe (2, 2) band is observed at ν0 = 13639.836(4) cm−1, only adifference of 2 cm−1 from the band origin of the (0, 1). This

6

deviation from typical vibrational spacings indicates that theupper state electronic potential is heavily perturbed. Similarly,the Π1[14.22]← 3∆2 (0, 0) band fit residuals deviate from thestandard form above J = 50. Here, no other vibrational levelshave been observed; the (0, 1) band is obscured by the strongΠ1[12.99]← 3∆2 (1, 0), and the (1, 1) and (2, 2) are not ob-served.

Four additional ThF+ electronic states were observed butnot conclusively assigned. In all cases, the lower electronicstate B values are not consistent with the states observed byBarker et al. [17]. This suggests that Te > 4,000 cm−1,which in turn suggests that this population is strongly superther-mal. Franck-Condon factors indicate that the band at ν0 =

10702.830(4) cm−1 is a vibrational (0, 0) and the two bandsat ν0 = 10085.683(7) and 10050.200(7) cm−1 are the (0, 1) and(1, 2), respectively. A third band was observed at ν0 ≈ 10,014cm−1, consistent with the location and intensity of the (2, 3),but was not fit. Each transition has two of each P-, Q-, andR-branch, again indicating Λ-doubling in each electronic state.The lower electronic state exhibits J′′4-scaling Λ-doubling witha strong vibrational dependence: k′′D = 34.0(7) × 10−9 cm−1

in the v′′ = 0, and it decreases to 14(1) × 10−9 cm−1 in thev′′ = 1 and 5.5(3) × 10−9 cm−1 in the v′′ = 2. Low-J line in-tensities in the (0, 0) and (0, 1) bands reveal that Ω′′ < Ω′.R(1) is absent and R(2) is present. The magnitude and scal-ing of the Λ-doubling, the lower value of B, and the low-J lines suggest that the lower electronic state may be a 3Π2state. With the vibrational assignment we can calculate con-stants for the electronic states. For the lower state, we obtainB′′e = 0.23402(5) cm−1, ω′′e = 619.8(2) cm−1, ωeχ

′′e = 1.34(9)

cm−1, and α′′e = 0.74(4) × 10−4 cm−1. For the upper statewe measure: B′e = 0.22363(5) cm−1, ω′e = 593.1(4) cm−1,ωeχ

′e = 1.70(9) cm−1, and α′e = 0.89(4) × 10−4 cm−1.

A pair of transitions was observed at ν0 = 12721.532(7) cm−1

and ν0 = 12661.053(7) cm−1. Each band has a single P- and R-branch and a strong Q-branch. k and kD both fit to 0, indicatingvery small Λ-doubling in one or both of the states. Due to spec-tral congestion and the relatively low intensity of the band, nolow-J lines were observed. The intensity of the bands is consis-tent with vibrational quantum numbers of ∆v = +1 or ∆v = −1.A third band, likely the (v′ + 2, v′′ + 2), can be barely resolvedat ν0 ≈ 12,600 cm−1, but it is too weak to fit. The (0, 0) istypically the strongest band, but is not immediately observedwhen searching within a reasonable range of ωe values; it islikely obscured by the presence of stronger bands in these re-gions. Without a definitive vibrational assignment we cannotconclusively determine Be, ωe, or ωeχe.

3.3. ThO+

We have also fit five bands that do not originate from ThF+.Based on the rotational constants, these bands appear to be orig-inating from two different electronic states, separated by about582 cm−1, coupling to a common upper state. The fit rotationalconstants are nearly 1.5 times larger than those for ThF+ values:B ≈ 0.34 cm−1, compared to B(ThF+) ≈ 0.24 cm−1. Addition-ally, the bands must be fit with half-integer J, indicative of amolecule with a spin doublet. ThO+, experimentally studied

by Goncharov et al. in 2006 [39], seems a promising candi-date. However, our measured constants, given in Table 4, donot form a fully consistent picture with the values measured byGoncharov et al., so we warn that our identification of ThO+ asthe unknown species is tentative.

ThO+ bands were fit to the form given in Eqn. 1, exceptwith J half-integer. All five transitions fit well out to J = 100with no Λ-doubling terms, i.e. k = kD = 0. The vibrationalassignment is tentative and is based on overall band intensitiesas predicted by Franck-Condon factors. The electronic statewith B0 = 0.33984(2) cm−1 is consistent with the 2∆3/2 state ofThO+. B′ = 0.32610(1) cm−1 is a reasonable value for a higher-lying electronic state. However, the third electronic state, withB0 = 0.33900(2) cm−1, is only consistent with a higher-lying2Σ+ or 2∆5/2 vibrational level. The upper electronic state ineach group of transitions appears to be identical; B is consis-tent to within 0.00002 cm−1. If the upper state is the same, thetwo lower states must be separated in energy by 12146.812(1)cm−1 - 11564.329(1) cm−1 = 582.483(2) cm−1. However, notwo measured states of ThO+ [39] are separated by this energy.Because these states had not been observed by Goncharov etal. [39], it is possible that the ThO+ population distribution inthe discharge is superthermal, resulting in the observation ofhigh-lying electronic states. Finally, we observed a ThO+ char-acteristic band head near 9800 cm−1, but it was sufficiently outof range of our cw Ti:Sapph that we did not scan enough of theband to attempt a fit.

4. Conclusion

We have determined that X 3∆1 is the electronic ground stateof ThF+, and a 1Σ+ is the first excited state, located 314 cm−1

above the ground state. This was assigned by observing twoΩ = 0+ states that couple to both X and a, at Te = 10,472and 14,296 cm−1. In an eEDM experiment it may be beneficialto first selectively ionize neutral ThF into the ThF+ 1Σ+ state,then transfer population to the ground state to obtain a purepopulation in a single Zeeman, Stark, and hyperfine sublevel[12]. Either Ω = 0+ state can be used to transfer populationbetween the two states. Transitions to the Ω = 0+[14.29] stateare nearly a factor of 10 stronger in relative intensity than to theΩ = 0+[10.47] state; however, the weaker transition is notablyon par with the transition strength to the 3Π0+ state observed inHfF+, which is already strong enough for coherent populationtransfer [21].

For the eEDM experiment, the Λ-doubling in 3∆1 is im-portant for setting the voltage required to fully polarize themolecule. The Λ-doubling in the ThF+ 3∆1 state is about 7times larger than in the HfF+ 3∆1 state [21]. The larger valuecan be explained by the higher density of states at low energy.The 3∆1 Λ-doubling can arise via L-uncoupling with a nearby3Π0 state, which can in turn couple to a Σ-state via a spin-orbitinteraction [40, 41]. Barker et al. observed a 3Π0 state at E =

3,395 cm−1 [17], three times closer in energy to 3∆1 than it isin HfF+. This would in turn lower the energy of 3Π1, anotherpotential interaction pathway, relative to HfF+. Additionally, it

7

is not unreasonable to expect a lower-lying Σ state to accountfor the remaining difference.

In addition, we have measured and assigned rotational andvibrational constants of the 3∆2 and 3∆3 states, and measuredrotational constants, vibrational constants, and electronic ener-gies for several intermediate electronic states. These values willprovide a benchmark for ab initio calculations. Derived con-stants for ThF+ electronic states are given in Table 2. A com-parison to other ThF+ experimental and theory results is givenin Table 3. Other than ambiguity in the ground state assign-ment, excellent agreement is seen between this work, Barkeret al. [17], and theory calculations from Denis et al. [15] andSkripnikov et al. [16].

Future directions for a ThF+ eEDM experiment will in-clude ionization spectroscopy of neutral ThF to determine ef-ficient ion creation pathways. The ThF dissociation energyis greater than the ionization energy [17, 32], which may al-low for greater flexibility in highly-efficient ion creation. Oneprospect is to prepare the neutral molecule in a core nonpen-etrating high-l Rydberg state, where it can vibrationally auto-ionize without perturbing the ion core [42, 43, 44]. Addition-ally, resonance-enhanced multiphoton dissociation (REMPD)spectroscopy will be performed on ThF+. Laser-induced flu-orescence (LIF) spectroscopy had previously been used as theprimary detection method on trapped HfF+ [45], but REMPDyielded a roughly 100-times improvement in count rate [46].We expect to use REMPD as our primary detection scheme intrapped ThF+.

In conclusion, we have demonstrated that comb-vms is apowerful tool for high-resolution extensive survey spectroscopyof molecular ions, in this case ThF+. Our wavelength coveragespans the 696 - 900 nm region, covering 3,260 cm−1 of con-tinuous spectra. Additional spectral regions in the visible andnear-IR could be accessed with additional broadening and a dif-ferent VIPA etalon.

Acknowledgements. Funding was provided by the MarsicoFoundation, NIST, and the NSF Physics Frontier Center at JILA(Grant No. 1125844). We thank M. Heaven, A. Titov, L. Skrip-nikov, A. Petrov, T. Fleig, R. Field, W. Cairncross, and M. Graufor useful discussions. Commercial products referenced in thiswork are not endorsed by NIST and are for the purposes oftechnical communication only.

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[47] T. Fleig, private communication (2015).

9

Table1:Fitted

constantsforobserved

transitionsinT

hF+

incm−

1.Quoted

uncertaintiesare

statisticalandare

atthe95%

confidencelevel.C

onstantslisted

arethe

rotationalconstant,B,centrifugaldistortion,D

,andgeneric

Λ-doubling

constants,k,andk

D,proportionalto

J(J+

1)and

J2(J

+1) 2,respectively.

Statisticalerrorson

bandorigins

aretypically

<90

MH

z,butanadditional150

MH

zis

addedto

includew

avemetererror.

ν0

B′′

B′

D′′

D′

k′′

k′

kD

[10−

7][10−

7][10−

4][10−

4][10−

9]

Ω=

0−←

3∆1

(0,2)13286.716(7)

0.24059(3)0.22898(3)

1.29(5)1.34(5)

-0.887(6)–

0Ω

=0−←

3∆1

(1,3)13231.099(8)

0.23960(4)0.22805(4)

1.33(9)1.37(9)

-0.90(1)–

0Ω

=0−←

3∆1

(0,1)13935.996(7)

0.24161(2)0.22899(2)

1.32(2)1.36(2)

-0.876(3)–

0Ω

=0−←

3∆1

(1,2)13876.539(8)

0.24061(3)0.22804(3)

1.31(5)1.36(5)

-0.882(6)–

0Ω

=0−←

3∆1

(2,3)13817.279(8)

0.23962(3)0.22711(3)

1.35(3)1.39(3)

-0.882(5)–

0Ω

=0

+[10.47]←

3∆1

(0,0)‡

10471.889(7)0.24264(3)

0.23534(3)1.30(4)

1.33(4)0.869(7)

–0

Ω=

0+[10

.47]←

3∆1

(1,1)‡

10441.560(8)0.24163(6)

0.23437(6)1.40(13)

1.35(13)0.874(13)

–0

Ω=

0+[14

.29]←

3∆1

(0,1)‡

13642.867(7)0.24161(5)

0.23194(5)1.37(13)

2.33(13)0.892(12)

–0

Ω=

0+[14

.29]←

3∆1

(0,0)‡

14295.998(8)0.24259(5)

0.23187(5)1.27(15)

2.23(15)0.884(14)

–0

Ω=

0+[14

.29]←

3∆1

(2,2)‡

13639.836(9)0.24064(7)

0.22998(7)1.34(13)

1.96(13)0.891(13)

–0

Ω=

0+[10

.47]←

1Σ+

(0,0)‡

10157.607(7)0.24542(3)

0.23534(3)+

1.35(4)1.33(4)

+0

00

Ω=

0+[14

.29]←

1Σ+

(0,1)‡

13328.320(8)0.24433(3)

0.23194(5)*1.46(13)

2.33(13)*0

00

Ω=

0+[14

.29]←

1Σ+

(0,0)‡

13981.718(8)0.24545(5)

0.23194(5)*1.42(13)

2.33(13)*0

00

Π1 [12

.99]←

3∆2

(0,1)11288.659(7)

0.24183(4)0.23001(3)

1.34(10)1.4

1(9)0

8.71(2)1.3(5)

Π1 [12

.99]←

3∆2

(0,0)11941.777(6)

0.24289(1)0.23005(1)

1.37(1)1.44(1)

08.70(1)

0.97(7)Π

1 [12.99]←

3∆2

(1,0)12519.028(7)

0.24287(2)0.22918(2)

1.35(3)1.43(3)

08.85(1)

2.1(2)Π

1 [12.99]←

3∆2

(2,1)§

12433.658(7)0.24179(3)

0.22853(3)1.29(6)

1.42(6)0

9.63(3)7.7(8)

Π1 [12

.99]←

3∆2

(3,2)12352.579(7)

0.24083(4)0.22775(4)

1.37(5)1.44(5)

09.05(3)

10.1(7)Π

1 [14.22]←

3∆2

(0,0)13171.134(7)

0.24284(4)0.23101(4)

1.29(12)1.08(12)

0-11.84(4)

-30(2)3Π

2←

3∆3

(0,1)‡

12877.103(7)0.24236(4)

0.22799(4)1.32(11)

1.75(11)0

-0.046(37)74(2)

3Π2←

3∆3

(1,2)12802.967(8)

0.24141(5)0.22675(5)

1.46(8)1.42(8)

0-0.22(3)

41(9)3Π

2←

3∆3

(0,0)‡

13532.407(7)0.24337(4)

0.22797(4)1.28(17)

1.68(17)0

-0.070(48)74(4)

?(0,1)

§10085.684(7)

0.23295(3)0.22322(3)

1.37(7)1.31(7)

0-0.08(2)

-14.0(7)†

?(1,2)

10050.200(7)0.23209(2)

0.22229(2)1.35(3)

1.32(3)0

-0.09(2)-5.5(3)

†

?(0,0)

‡10702.830(9)

0.23365(3)0.22323(3)

1.30(5)1.31(5)

0-0.19(2)

-34.0(7)†

?(v′,v′′)

12721.532(8)0.23466(5)

0.22308(5)1.31(7)

1.32(7)0

00

?(v′+

1,v′′

+1)

12661.053(8)0.23361(6)

0.22203(6)1.16(13)

1.17(13)0

00

k′D

unlessotherw

isenoted.

†k′′D

.+

Values

fixedto

B′and

D′from

theΩ

=0

+[10.47]←

3∆1

(0,0)transition.*

Values

fixedto

B′and

D′from

theΩ

=0

+[14.29]←

3∆1

(0,1)transition.‡

FittoJm

ax=

50due

toperturbations.

§Fitto

Jmax

=60.

10

Table 2: Derived constants for observed states in ThF+ in cm−1. T0 is the origin of the v = 0 level relative to the v = 0 levelof X 3∆1. T0 errors are directly measured in transitions from 3∆1 and 1Σ+; otherwise, error bars are adapted from [17]. Vibrationalconstants are calculated assuming a Morse potential unless otherwise noted. Quoted errors are 95%.

State T0 Te Be ωe ωeχe αe [10−3]3∆1 0 0 0.24311(7) 656.96(1)+ 1.920(3)+ 1.00(4)1Σ+ 314.282(7) 314.0(2) 0.24601(8) 657.90(32) 2.26(16) 1.12(6)3∆2 1052.5(1.0)† 1052.5(1.0)‡ 0.24342(2) 657.38(22) 2.13(10) 1.06(4)3∆3 3150(30)† 3149(30)‡ 0.24388(7) 659.31(29) 2.00(15) 1.01(6)Ω = 0+[10.47] 10471.889(7) 10487.1(1) 0.23583(5) 626.67(56) 1.88(17) 0.97(7)Π1[12.99] 12994.3(1.0)‡ 13032.6(1.0)‡ 0.23049(2) 580.33(10) 1.54(5) 0.87(2)Π1[14.22] 14223.6(1.0)‡ – 0.23101(4)* – – –Ω = 0+[14.29] 14295.967(8) – 0.23187(5)* – – –Ω = 0− 14589.09(2) 14620.81(1) 0.22947(3) 593.467(8)+ 1.822(3)+ 0.95(4)3Π2 16682(30)‡ 16719(30)‡ 0.22858(7) 582.04(75) 2.44(18) 1.22(6)+ Calculated without assuming a Morse potential.* B0.† Values and error bars from [17].‡ Calculated using 3∆2 and 3∆3 energies and errors from [17].

Morse potential assumption breaks down for v′ = 2 and 3.

Table 3: Comparison of low-lying states. Measured Te, Be, and ωe values are com-pared from this work to measured values from Barker et al. [17], and to theory calcula-tions from Denis et al. [15] and Skripnikov et al. [16]. Quoted uncertainties are 95%.

State Constant This Work [17] [15] [16]3∆1 Te 0 315(1) 0 0

Be 0.24311(7) 0.246(10) 0.2439 0.2438ωe 656.96(1) 658.3(2.0) 667.3 658.4

1Σ+ Te 314.0(2) 0 318.99 448‡

Be 0.24601(8) 0.245(10) 0.2464 –ωe 657.9(3) 656.8(2.0) 670.8 –

3∆2 Te – 1052.5(1.0) 1038.94 1186‡

Be 0.24342(2) 0.243(10) 0.2438† –ωe 657.38(22) 656.5(2.0) 667.0† –

3∆3 Te – 3150(30) 3161.99 3193‡

Be 0.24388(7) – 0.2443† –ωe 659.33(26) 665(30)* 669.0† –

* ∆G1/2.† Stefan Knecht, unpublished results [47].‡ Preliminary data for vertical transitions at Re = 3.75 a.u. [19]. The accuracy of

the calculations has since been improved.

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Table 4: Fitted constants for observed transitions in ThO+. Errors are statistical only andat the 95% confidence level. Bands were fit with half-integer J. No Λ-doubling terms weremeasured. Statistical errors on band origins are < 60 MHz, but an additional 150 MHz is addedto include wavemeter error. No electronic states were able to be conclusively assigned.

ν0 B′′ B′ D′′ D′

[10−7] [10−7]

? (0, 0) 12146.812(6) 0.33984(2) 0.32610(1) 1.91(2) 1.89(2)? (1, 1) 12091.106(7) 0.33843(4) 0.32469(4) 1.78(7) 1.77(6)? (1, 0) 13006.858(7) 0.33973(4) 0.32465(4) 1.6(1) 1.6(1)? (0, 0) 11564.329(6) 0.33900(2) 0.32612(2) 1.86(2) 1.87(2)? (1, 1) 11510.656(7) 0.33767(3) 0.32480(3) 1.86(5) 1.87(5)

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